Proteins are a diverse class of biomolecules responsible for wide-ranging cellular functions, from catalyzing reactions to recognizing pathogens. The ability to evolve proteins rapidly and inexpensively toward improved properties is a common objective for protein engineers. Powerful high-throughput methods like fluorescent activated cell sorting and next-generation sequencing have dramatically improved directed evolution experiments. However, it is unclear how to best leverage these data to characterize protein fitness landscapes more completely and identify lead candidates. In this work, we develop a simple yet powerful framework to improve protein optimization by predicting continuous protein properties from simple directed evolution experiments using interpretable, linear machine learning models. Importantly, we find that these models, which use data from simple but imprecise experimental estimates of protein fitness, have predictive capabilities that approach more precise but expensive data. Evaluated across five diverse protein engineering tasks, continuous properties are consistently predicted from readily available deep sequencing data, demonstrating that protein fitness space can be reasonably well modeled by linear relationships among sequence mutations. To prospectively test the utility of this approach, we generated a library of stapled peptides and applied the framework to predict affinity and specificity from simple cell sorting data. We then coupled integer linear programming, a method to optimize protein fitness from linear weights, with mutation scores from machine learning to identify variants in unseen sequence space that have improved and co-optimal properties. This approach represents a versatile tool for improved analysis and identification of protein variants across many domains of protein engineering.